Lower Bounds on Sparse Spanners, Emulators, and Diameter-reducing shortcuts

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Omega(n^{1/6}), and that any O(m)-size shortcut set cannot bring it below Omega(n^{1/11}). These improve Hesse\u27s [Hesse, 2003] lower bound of Omega(n^{1/17}). By combining these constructions with Abboud and Bodwin\u27s [Abboud and Bodwin, 2017] edge-splitting technique, we get additive stretch lower bounds of +Omega(n^{1/13}) for O(n)-size spanners and +Omega(n^{1/18}) for O(n)-size emulators. These improve Abboud and Bodwin\u27s +Omega(n^{1/22}) lower bounds

Towards Bypassing Lower Bounds for Graph Shortcuts

For a given (possibly directed) graph G, a hopset (a.k.a. shortcut set) is a (small) set of edges whose addition reduces the graph diameter while preserving desired properties from the given graph G, such as, reachability and shortest-path distances. The key objective is in optimizing the tradeoff between the achieved diameter and the size of the shortcut set (possibly also, the distance distortion). Despite the centrality of these objects and their thorough study over the years, there are still significant gaps between the known upper and lower bound results. A common property shared by almost all known shortcut lower bounds is that they hold for the seemingly simpler task of reducing the diameter of the given graph, D_G, by a constant additive term, in fact, even by just one! We denote such restricted structures by (D_G-1)-diameter hopsets. In this paper we show that this relaxation can be leveraged to narrow the current gaps, and in certain cases to also bypass the known lower bound results, when restricting to sparse graphs (with O(n) edges): - {Hopsets for Directed Weighted Sparse Graphs.} For every n-vertex directed and weighted sparse graph G with D_G ? n^{1/4}, one can compute an exact (D_G-1)-diameter hopset of linear size. Combining this with known lower bound results for dense graphs, we get a separation between dense and sparse graphs, hence shortcutting sparse graphs is provably easier. For reachability hopsets, we can provide (D_G-1)-diameter hopsets of linear size, for sparse DAGs, already for D_G ? n^{1/5}. This should be compared with the diameter bound of O?(n^{1/3}) [Kogan and Parter, SODA 2022], and the lower bound of D_G = n^{1/6} by [Huang and Pettie, {SIAM} J. Discret. Math. 2018]. - {Additive Hopsets for Undirected and Unweighted Graphs.} We show a construction of +24 additive (D_G-1)-diameter hopsets with linear number of edges for D_G ? n^{1/12} for sparse graphs. This bypasses the current lower bound of D_G = n^{1/6} obtained for exact (D_G-1)-diameter hopset by [HP\u2718]. For general graphs, the bound becomes D_G ? n^{1/6} which matches the lower bound of exact (D_G-1) hopsets implied by [HP\u2718]. We also provide new additive D-diameter hopsets with linear size, for any given diameter D. Altogether, we show that the current lower bounds can be bypassed by restricting to sparse graphs (with O(n) edges). Moreover, the gaps are narrowed significantly for any graph by allowing for a constant additive stretch

Dynamic tree shortcut with constant degree

LNCS v.9188 entitled: Computing and Combinatorics: 21st International Conference, COCOON 2015, Beijing, China, August 4-6, 2015, ProceedingsGiven a rooted tree with n nodes, the tree shortcut problem is to add a set of shortcut edges to the tree such that the shortest path from each node to any of its ancestors is of length O(log n) and the degree increment of each node is constant. We consider in this paper the dynamic version of the problem, which supports node insertion and deletion. For insertion, a node can be inserted as a leaf node or an internal node by sub-dividing an existing edge. For deletion, a leaf node can be deleted, or an internal node can be merged with its single child. We propose an algorithm that maintains a set of shortcut edges in O(log n) time for an insertion or deletion.postprin

Simpler and Higher Lower Bounds for Shortcut Sets

We provide a variety of lower bounds for the well-known shortcut set problem: how much can one decrease the diameter of a directed graph on $n$ vertices and $m$ edges by adding $O(n)$ or $O(m)$ of shortcuts from the transitive closure of the graph. Our results are based on a vast simplification of the recent construction of Bodwin and Hoppenworth [FOCS 2023] which was used to show an $\widetilde{\Omega}(n^{1/4})$ lower bound for the $O(n)$-sized shortcut set problem. We highlight that our simplification completely removes the use of the convex sets by B\'ar\'any and Larman [Math. Ann. 1998] used in all previous lower bound constructions. Our simplification also removes the need for randomness and further removes some log factors. This allows us to generalize the construction to higher dimensions, which in turn can be used to show the following results. For $O(m)$-sized shortcut sets, we show an $\Omega(n^{1/5})$ lower bound, improving on the previous best $\Omega(n^{1/8})$ lower bound. For all $\varepsilon > 0$, we show that there exists a $\delta > 0$ such that there are $n$-vertex $O(n)$-edge graphs $G$ where adding any shortcut set of size $O(n^{2-\varepsilon})$ keeps the diameter of $G$ at $\Omega(n^\delta)$. This improves the sparsity of the constructed graph compared to a known similar result by Hesse [SODA 2003]. We also consider the sourcewise setting for shortcut sets: given a graph $G=(V,E)$, a set $S\subseteq V$, how much can we decrease the sourcewise diameter of $G$, $\max_{(s, v) \in S \times V, \text{dist}(s, v) < \infty} \text{dist}(s,v)$ by adding a set of edges $H$ from the transitive closure of $G$? We show that for any integer $d \ge 2$, there exists a graph $G=(V, E)$ on $n$ vertices and $S \subseteq V$ with $|S| = \widetilde{\Theta}(n^{3/(d+3)})$, such that when adding $O(n)$ or $O(m)$ shortcuts, the sourcewise diameter is $\widetilde{\Omega}(|S|^{1/3})$.Comment: To appear in SODA 2024. Abstract shortened to fit arXiv requirement

Are there graphs whose shortest path structure requires large edge weights?

The aspect ratio of a weighted graph $G$ is the ratio of its maximum edge weight to its minimum edge weight. Aspect ratio commonly arises as a complexity measure in graph algorithms, especially related to the computation of shortest paths. Popular paradigms are to interpolate between the settings of weighted and unweighted input graphs by incurring a dependence on aspect ratio, or by simply restricting attention to input graphs of low aspect ratio. This paper studies the effects of these paradigms, investigating whether graphs of low aspect ratio have more structured shortest paths than graphs in general. In particular, we raise the question of whether one can generally take a graph of large aspect ratio and \emph{reweight} its edges, to obtain a graph with bounded aspect ratio while preserving the structure of its shortest paths. Our findings are: - Every weighted DAG on $n$ nodes has a shortest-paths preserving graph of aspect ratio $O(n)$. A simple lower bound shows that this is tight. - The previous result does not extend to general directed or undirected graphs; in fact, the answer turns out to be exponential in these settings. In particular, we construct directed and undirected $n$-node graphs for which any shortest-paths preserving graph has aspect ratio $2^{\Omega(n)}$. We also consider the approximate version of this problem, where the goal is for shortest paths in $H$ to correspond to approximate shortest paths in $G$. We show that our exponential lower bounds extend even to this setting. We also show that in a closely related model, where approximate shortest paths in $H$ must also correspond to approximate shortest paths in $G$, even DAGs require exponential aspect ratio

Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product

We obtain improved lower bounds for additive spanners, additive emulators, and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of edges that when added to a directed graph, decreases its diameter. The previous best known lower bounds for these three structures are given by Huang and Pettie [SWAT 2018]. For $O(n)$-sized spanners, we improve the lower bound on the additive stretch from $\Omega(n^{1/11})$ to $\Omega(n^{2/21})$. For $O(n)$-sized emulators, we improve the lower bound on the additive stretch from $\Omega(n^{1/18})$ to $\Omega(n^{2/29})$. For $O(m)$-sized shortcut sets, we improve the lower bound on the graph diameter from $\Omega(n^{1/11})$ to $\Omega(n^{1/8})$. Our key technical contribution, which is the basis of all of our bounds, is an improvement of a graph product known as an alternation product.Comment: To appear in SODA 202

Closing the Gap Between Directed Hopsets and Shortcut Sets

For an n-vertex directed graph $G = (V,E)$, a $\beta$-\emph{shortcut set} $H$ is a set of additional edges $H \subseteq V \times V$ such that $G \cup H$ has the same transitive closure as $G$, and for every pair $u,v \in V$, there is a $uv$-path in $G \cup H$ with at most $\beta$ edges. A natural generalization of shortcut sets to distances is a $(\beta,\epsilon)$-\emph{hopset} $H \subseteq V \times V$, where the requirement is that $H$ and $G \cup H$ have the same shortest-path distances, and for every $u,v \in V$, there is a $(1+\epsilon)$-approximate shortest path in $G \cup H$ with at most $\beta$ edges. There is a large literature on the tradeoff between the size of a shortcut set / hopset and the value of $\beta$. We highlight the most natural point on this tradeoff: what is the minimum value of $\beta$, such that for any graph $G$, there exists a $\beta$-shortcut set (or a $(\beta,\epsilon)$-hopset) with $O(n)$ edges? Not only is this a natural structural question in its own right, but shortcuts sets / hopsets form the core of many distributed, parallel, and dynamic algorithms for reachability / shortest paths. Until very recently the best known upper bound was a folklore construction showing $\beta = O(n^{1/2})$, but in a breakthrough result Kogan and Parter [SODA 2022] improve this to $\beta = \tilde{O}(n^{1/3})$ for shortcut sets and $\tilde{O}(n^{2/5})$ for hopsets. Our result is to close the gap between shortcut sets and hopsets. That is, we show that for any graph $G$ and any fixed $\epsilon$ there is a $(\tilde{O}(n^{1/3}),\epsilon)$ hopset with $O(n)$ edges. More generally, we achieve a smooth tradeoff between hopset size and $\beta$ which exactly matches the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors). Using a very recent black-box reduction of Kogan and Parter, our new hopset implies improved bounds for approximate distance preservers.Comment: Abstract shortened to meet arXiv requirements, v2: fixed a typ

Folklore Sampling is Optimal for Exact Hopsets: Confirming the n\sqrt{n} Barrier

For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+\epsilon)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetilde{\Omega}(n^{1/2})$. This improves on the previous lower bound of $\widetilde{\Omega}(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetilde{\Omega}(n^{1/4})$. This improves on the previous lower bound of $\Omega(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$